Sounds like a faulty thermostat. If you are willing to relace it, do a search and find their customer service number and give them a call, there is a fair chance they will be more than happy to send a new one free of charge. Most of the time the t-stats only requires removal of a few screws to remove a cover to gain access, the wires are attached with spade connectors that come right off and are just as easily put back on. Even with such a simple fix, mini fridges are cheaply made, considered non servicable and the manufacture's protocol is just to have faulty units destroyed.
If the customer service call does not work, take it back to the place of purchase and ask for a replacement. If they are a good business, should be more than happy to do this, the manufacture issues automatic credit to them on any faulty units. Goes back to what I said earlier. If your model has been out of production for sometime, it would be a reasonable assumption you got the usefulness out of the unit and grounds to decline an exchange. Regardless, it is still worth a try.